I was listening to the radio the other day in the car and there was some program on about poetry which I was not really paying much attention to, but what caught my ear, so to speak, was mention that Homer is an anonymous figure, that they really don't know who he was.

Can anyone elaborate on this?

As far as I know, speculations on the identity of Homer vary widely; people have speculated that he was a soldier (doesn't he make it sufficiently clear he knows nothing about warfare??), a doctor, not a man but a school, blind, a sadomasochist (the erotic mass hanging of the twelve treacherous maidens), having committed suicide, that the writer of the Odyssey was not the same as that of the Illiad (and that it was a woman in Sicily), etc...

Scholars know with good certainty that Homer did not write down his works himself.  He was blind and traveled around as a "bard," telling his stories and adding to them as he went.  The original ideas were his and were passed down in an oral tradition ; it wasn't until later that they were actually recorded in a physical record.